Having problems with the bevel tool
So what's happening here is your object has a fair number of double vertices. Blender has a tool for removing doubles which can be found by pressing W in edit mode with everything selected. This does a decent job, but as a result created some triangles and n-gons (faces with more than 4 sides) which you definitely don't want. As well, it doesn't fix the original problem.

The reason these double vertices are a problem is because beveling adds more loops and then slides them on the edges connected to the vertices you had selected. Double vertices/faces/edges results in some of the connecting edges have a length of 0 and thus no direction. So blender doesn't know which way to slide the loops created by the bevel operation and you get the messy results you see.

To fix the problem, you have to remove the double vertices as well as clean up some extra edge loops and messy faces that are left over. It's definitely worth a try to fix this yourself as you will learn much better by solving the problem yourself. It can be pretty tricky to clean up a mesh though, so I uploaded a file that's been cleaned up and bevels properly.

One of the most common causes for double geometry, especially among beginners, is when you extrude something with E whether accidentally or intentionally and then immediately cancel the extrusion with Right click. This still extrudes the mesh, but leaves the extruded mesh directly on top of the original mesh, thus causing double geometry.